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The **Fresnel integrals***S*(*x*) and *C*(*x*) are two transcendental functions named after Augustin-Jean Fresnel that are used in optics and are closely related to the error function (erf). They arise in the description of near-field Fresnel diffraction phenomena and are defined through the following integral representations:

A **transcendental function** is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a **transcendental function** "transcends" algebra in that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, and root extraction.

**Augustin-Jean Fresnel** was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.

**Optics** is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.

- Definition
- Euler spiral
- Properties
- Limits as x approaches infinity
- Generalization
- Numerical approximation
- Applications
- See also
- References
- External links

The simultaneous parametric plot of *S*(*x*) and *C*(*x*) is the Euler spiral (also known as the Cornu spiral or clothoid). Recently, they have been used in the design of highways and other engineering projects.^{ [1] }

In mathematics, a **parametric equation** defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a **parametric representation** or **parameterization** of the object.

An **Euler spiral** is a curve whose curvature changes linearly with its curve length. Euler spirals are also commonly referred to as **spiros**, **clothoids**, or **Cornu spirals**.

The Fresnel integrals admit the following power series expansions that converge for all *x*:

Some authors, including Abramowitz and Stegun, (eqs 7.3.1 – 7.3.2) use for the argument of the integrals defining *S*(*x*) and *C*(*x*). This changes their limits at infinity from to and the arc length for the first spiral turn to 2 (at ), all smaller by a factor . The alternative functions are also called *Normalized Fresnel integrals*.

* Abramowitz and Stegun* (

The **Euler spiral **, also known as **Cornu spiral** or **clothoid**, is the curve generated by a parametric plot of against . The Cornu spiral was created by Marie Alfred Cornu as a nomogram for diffraction computations in science and engineering.

In mathematics, a **spiral** is a curve which emanates from a point, moving farther away as it revolves around the point.

**Marie Alfred Cornu** was a French physicist. The French generally refer to him as Alfred Cornu.

A **nomogram**, also called a **nomograph**, **alignment chart** or **abaque**, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d’Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision. Nomograms use a parallel coordinate system invented by d'Ocagne rather than standard Cartesian coordinates.

From the definitions of Fresnel integrals, the infinitesimals and are thus:

In mathematics, **infinitesimals** are things so small that there is no way to measure them. The insight with exploiting infinitesimals was that entities could still retain certain specific properties, such as angle or slope, even though these entities were quantitatively small. The word *infinitesimal* comes from a 17th-century Modern Latin coinage *infinitesimus*, which originally referred to the "infinity-th" item in a sequence. Infinitesimals are a basic ingredient in the procedures of infinitesimal calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity. In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective, "infinitesimal" means "extremely small". To give it a meaning, it usually must be compared to another infinitesimal object in the same context. Infinitely many infinitesimals are summed to produce an integral.

Thus the length of the spiral measured from the origin can be expressed as

That is, the parameter is the curve length measured from the origin , and the Euler spiral has infinite length. The vector also expresses the unit tangent vector along the spiral, giving . Since *t* is the curve length, the curvature can be expressed as

And the rate of change of curvature with respect to the curve length is

An Euler spiral has the property that its curvature at any point is proportional to the distance along the spiral, measured from the origin. This property makes it useful as a transition curve in highway and railway engineering: If a vehicle follows the spiral at unit speed, the parameter in the above derivatives also represents the time. That is, a vehicle following the spiral at constant speed will have a constant rate of angular acceleration.

Sections from Euler spirals are commonly incorporated into the shape of roller-coaster loops to make what are known as clothoid loops.

*C*(*x*) and*S*(*x*) are odd functions of*x*.- Asymptotics of the Fresnel integrals as are given by the formulas:

- Using the power series expansions above, the Fresnel integrals can be extended to the domain of complex numbers, and they become analytic functions of a complex variable.
- The Fresnel integrals can be expressed using the error function as follows:
^{ [2] }

- or

*C*and*S*are entire functions.

The integrals defining *C*(*x*) and *S*(*x*) cannot be evaluated in the closed form in terms of elementary functions, except in special cases. The limits of these functions as *x* goes to infinity are known:

The limits of *C* and *S* as the argument tends to infinity can be found by the methods of complex analysis. This uses the contour integral of the function

around the boundary of the sector-shaped region in the complex plane formed by the positive *x*-axis, the bisector of the first quadrant *y* = *x* with *x* ≥ 0, and a circular arc of radius *R* centered at the origin.

As *R* goes to infinity, the integral along the circular arc tends to 0

where polar coordinates were used and Jordan's inequality was utilised for the second inequality. The integral along the real axis tends to the half Gaussian integral

Note too that because the integrand is an entire function on the complex plane, its integral along the whole contour is zero. Overall, we must have

where denotes the bisector of the first quadrant, as in the diagram. To evaluate the right hand side, parametrize the bisector as

where r ranges from 0 to . Note that the square of this expression is just . Therefore, substitution gives the right hand side as

Using Euler's formula to take real and imaginary parts of gives this as

where we have written to emphasize that the original Gaussian integral's value is completely real with zero imaginary part. Letting and then equating real and imaginary parts produces the following system of two equations in the two unknowns :

Solving this for and gives the desired result.

The integral

is a confluent hypergeometric function and also an incomplete gamma function ^{ [3] }

which reduces to Fresnel integrals if real or imaginary parts are taken:

- .

The leading term in the asymptotic expansion is

and therefore

For *m* = 0, the imaginary part of this equation in particular is

with the left-hand side converging for *a* > 1 and the right-hand side being its analytical extension to the whole plane less where lie the poles of .

The Kummer transformation of the confluent hypergeometric function is

with

For computation to arbitrary precision, the power series is suitable for small argument. For large argument, asymptotic expansions^{ [4] } converge faster. Continued fraction methods may also be used.^{ [5] }

For computation to particular target precision, other approximations have been developed. Cody^{ [6] } developed a set of efficient approximations based on rational functions that give relative errors down to 2×10^{−19}. A FORTRAN implementation of the Cody approximation that includes the values of the coefficients needed for implementation in other languages was published by van Snyder.^{ [7] } Boersma developed an approximation with error less than 1.6×10^{−9}.^{ [8] }

The Fresnel integrals were originally used in the calculation of the electromagnetic field intensity in an environment where light bends around opaque objects.^{ [9] } More recently, they have been used in the design of highways and railways, specifically their curvature transition zones, see track transition curve.^{ [1] } Other applications are roller coasters^{ [9] } or calculating the transitions on a velodrome track to allow rapid entry to the bends and gradual exit.^{[ citation needed ]}

In mathematics, the **arithmetic–geometric mean** (**AGM**) of two positive real numbers *x* and *y* is defined as follows:

**Bessel functions**, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions *y*(*x*) of Bessel's differential equation

In mathematics, the **error function** is a special function (non-elementary) of sigmoid shape that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:

Integration is the basic operation in integral calculus. While differentiation has easy rules by which the derivative of a complicated function can be found by differentiating its simpler component functions, integration does not, so tables of known integrals are often useful. This page lists some of the most common antiderivatives.

In mathematics, a **Gaussian function**, often simply referred to as a **Gaussian**, is a function of the form:

In the physical sciences, the **Airy function****Ai( x)** is a special function named after the British astronomer George Biddell Airy (1801–1892). The function Ai(

In mathematics, the **Clausen function**, introduced by Thomas Clausen (1832), is a transcendental, special function of a single variable. It can variously be expressed in the form of a definite integral, a trigonometric series, and various other special functions. It is intimately connected with the polylogarithm, inverse tangent integral, polygamma function, Riemann zeta function, Dirichlet eta function, and Dirichlet beta function.

In mathematics, the **Dawson function** or **Dawson integral** is either

The **Gaussian integral**, also known as the **Euler–Poisson integral**, is the integral of the Gaussian function *e*^{−x2} over the entire real line. It is named after the German mathematician Carl Friedrich Gauss. The integral is:

In the mathematical field of complex analysis, **contour integration** is a method of evaluating certain integrals along paths in the complex plane.

In mathematics, there are several integrals known as the **Dirichlet integral**, after the German mathematician Peter Gustav Lejeune Dirichlet.

In mathematics, the Fourier **sine and cosine transforms** are forms of the Fourier integral transform that do not use complex numbers. They are the forms originally used by Joseph Fourier and are still preferred in some applications, such as signal processing or statistics.

In complex analysis, **Jordan's lemma** is a result frequently used in conjunction with the residue theorem to evaluate contour integrals and improper integrals. It is named after the French mathematician Camille Jordan.

There are **common integrals in quantum field theory** that appear repeatedly. These integrals are all variations and generalizations of gaussian integrals to the complex plane and to multiple dimensions. Other integrals can be approximated by versions of the gaussian integral. Fourier integrals are also considered.

**Landen's transformation** is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.

In mathematics, and more precisely in analysis, the **Wallis integrals** constitute a family of integrals introduced by John Wallis.

In mathematics, a **Böhmer integral** is an integral introduced by Böhmer (1939) generalizing the Fresnel integrals.

- 1 2 James Stewart (2008).
*Calculus Early Transcendentals*. Cengage Learning EMEA. p. 383. ISBN 978-0-495-38273-7. - ↑ functions.wolfram.com, Fresnel integral S: Representations through equivalent functions and Fresnel integral C: Representations through equivalent functions. Note: Wolfram uses the Abramowitz & Stegun convention, which differs from the one in this article by factors of
- ↑ Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv: 1211.3963 [math.CA].
- ↑ Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals: Asymptotic expansions", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0521192255, MR 2723248 - ↑ Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007). "Section 6.8.1. Fresnel Integrals".
*Numerical Recipes: The Art of Scientific Computing*(3rd ed.). New York: Cambridge University Press. ISBN 978-0-521-88068-8. - ↑ Cody, William J (1968). "Chebyshev approximations for the Fresnel integrals" (PDF).
*Math. Comp*.**22**(102): 450–453. doi:10.1090/S0025-5718-68-99871-2. - ↑ van Snyder, W (December 1993). "Algorithm 723: Fresnel integrals".
*ACM Trans. Math. Softw*.**19**(4): 452–456. doi:10.1145/168173.168193. - ↑ Boersma, J. (1960). "Computation of Fresnel Integrals".
*Math. Comp*.**14**(72): 380. doi:10.1090/S0025-5718-1960-0121973-3. MR 0121973. - 1 2 Beatty, Thomas. "How to evaluate Fresnel Integrals" (PDF).
*FGCU MATH - SUMMER 2013*. Retrieved 27 July 2013.

- van Wijngaarden, A.; Scheen, W. L. (1949).
*Table of Fresnel Integrals*. Verhandl. Konink. Ned. Akad. Wetenschapen.**19**. - Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 7".
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables*. Applied Mathematics Series.**55**(Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 297. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. - Bulirsch, Roland (1967). "Numerical calculation of the sine, cosine and Fresnel integrals".
*Numer. Math*.**9**(5): 380–385. doi:10.1007/BF02162153. - Hangelbroek, R. J. (1967). "Numerical approximation of Fresnel integrals by means of Chebyshev polynomials".
*J. Eng. Math*.**1**(1): 37–50. Bibcode:1967JEnMa...1...37H. doi:10.1007/BF01793638. - Nave, R. (2002). "The Cornu spiral".
*(Uses πt²/2 instead of t².)* - Temme, N. M. (2010), "Error Functions, Dawson's and Fresnel Integrals", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.),
*NIST Handbook of Mathematical Functions*, Cambridge University Press, ISBN 978-0521192255, MR 2723248 - Alazah, Mohammad (2012). "Computing fresnel integrals via modified trapezium rules".
*Numerische Mathematik*.**128**(4): 635–661. arXiv: 1209.3451 . Bibcode:2012arXiv1209.3451A. doi:10.1007/s00211-014-0627-z. - Mathar, R. J. (2012). "Series Expansion of Generalized Fresnel Integrals". arXiv: 1211.3963 [math.CA].

- Cephes, free/open-source C++/C code to compute Fresnel integrals among other special functions. Used in SciPy and ALGLIB.
- Faddeeva Package, free/open-source C++/C code to compute complex error functions (from which the Fresnel integrals can be obtained), with wrappers for Matlab, Python, and other languages.
- Hazewinkel, Michiel, ed. (2001) [1994], "Fresnel integrals",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 - "Roller Coaster Loop Shapes". Archived from the original on September 23, 2008. Retrieved 2008-08-13.
- Weisstein, Eric W. "Fresnel Integrals".
*MathWorld*. - Weisstein, Eric W. "Cornu Spiral".
*MathWorld*.

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